Checking Moment Existence for an Exponential Lévy Price Model

Summary

The document asks whether a risky asset modeled as an exponential of a Lévy process is well defined and has finite first and second moments. Its Lévy measure combines negative jumps at integer sizes, weighted by a geometric sequence, with positive jumps governed by a tempered power-law density. The question is how these jump components, along with the process parameters, affect existence and finiteness of moments.

No derivation or answer is provided, so the conditions needed to establish the claims remain unresolved. In general, exponential moments depend on the Lévy measure's integrability against exponential functions, and the drift and Gaussian component also enter the moment calculation. The document supplies no parameter restrictions or numerical evidence, making it a prompt for applying Lévy-process moment criteria rather than a demonstrated result.

Key ideas

• The asset price is specified as an exponential transform of a Lévy process with a constant interest-rate term.
• The jump measure combines geometrically weighted negative integer jumps and a tempered positive-jump density.
• Finite price moments require checking exponential integrability of the Lévy measure.
• The document poses the moment-existence question but does not supply a derivation or parameter conditions.

Tags

# Simple question concerning Jump process (Lévy process) model for a risky actif price process


# Simple question concerning Jump process (Lévy process) model for a risky actif price process












Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is $$ \nu \left( dx\right) = A \sum_{n=1} ^{\infty} p^n \delta_{-n}\left( dx \right) + Bx^{\beta-1}\left( 1+x \right)^{-\alpha -\beta}e^{-\lambda x } \mathbf{1}_{\left ]0,+\infty \right[}\left( x\right)dx.$$

I'd like to know how to show that a price process $S_t = S_0 \exp\left( r t + X_t \right)$ of a risky actif under interest rate $r >0$ is well defined and admits first and second order moments.

Someone could help on it, please?

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